There Exist Non-trivial Pl Knots Whose Complements Are Homotopy Circles

We show that there exist non-trivial PL knots S n−2 ⊂ S n , n ≥ 5, whose complements have the homotopy type of circles. This is in contrast to the case of smooth, PL locally-flat, and topological locally-flat knots, for which it is known that if the complement has the homotopy type of a circle, then the knot is trivial. It is well-known that if the complement of a smooth, PL locally-flat, or topological locally-flat knot K ⊂ S n , K ∼ = S n−2 , n ≥ 5, has the homotopy type of a circle, then K is equivalent to the standard unknot in the appropriate category (see Stallings [8] for the topological case and Levine [3] and [5, §23] for the smooth and PL cases). We will show, however, that this does not hold in the PL category once the condition of local-flatness has been removed. In fact, we will construct PL knots for any n ≥ 5 that are locally-flat except at one point and whose complements are homotopy circles.